Chapter 2
Quadratic Operators and Quadratic Functional
Equation
M. Adam and S. Czerwik
Abstract In the first part of this paper, we consider some quadratic difference
operators (e.g., Lobaczewski difference operators) and quadratic-linear difference
operators (d’Alembert difference operators and quadratic difference operators) in
some special function spaces Xλ . We present results about boundedness and find
the norms of such operators. We also present new results about the quadratic functional equation. The second part is devoted to the so-called double quadratic difference property in the class of differentiable functions. As an application we prove
the stability result in the sense of Ulam–Hyers–Rassias for the quadratic functional
equation in a special class of differentiable functions.
Key words Quadratic, d’Alembert, and Lobaczewski difference operators ·
Xλ spaces · Quadratic functional equation · Stability
Mathematics Subject Classification 39B52 · 39B82 · 47H30
2.1 The Xλ and Xλ2 Spaces
We shall introduce the spaces Xλ and Xλ2 (see [7]). A. Bielecki also studied similar
spaces in [4] and applied them in the theory of differential equations.
Definition 2.1 Let X and Y be two normed vector spaces and λ ≥ 0. Define
Xλ := f : X → Y : f (x) ≤ Mf eλx , x ∈ X ,
Dedicated to Professor Themistocles M. Rassias on the occasion of his 60th birthday.
M. Adam
Department of Mathematics and Informatics, Higher School of Labour Safety Management in
Katowice, Bankowa 8, 40-007 Katowice, Poland
e-mail: [email protected]
S. Czerwik ()
Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice,
Poland
e-mail: [email protected]
P.M. Pardalos et al. (eds.), Nonlinear Analysis, Springer Optimization and Its
Applications 68, In Honor of Themistocles M. Rassias on the Occasion of his 60th Birthday,
DOI 10.1007/978-1-4614-3498-6_2, © Springer Science+Business Media, LLC 2012
13
14
M. Adam and S. Czerwik
where Mf is a real constant depending on f. Moreover, for f ∈ Xλ
f := sup e−λx f (x) .
(2.1)
x∈X
Let us note that the space Xλ with the norm (2.1) was considered by S. Czerwik
and K. Dłutek in [10]. It is easy to prove the following
Lemma 2.1 The space (Xλ , · ), where · is defined by (2.1), is a linear normed
space.
Definition 2.2 Let X and Y be two normed vector spaces and λ ≥ 0. Define
Xλ2 := g : X × X → Y : g(x, y) ≤ Mg eλ(x+y) , x, y ∈ X ,
where Mg is a real constant depending on g. Moreover, for g ∈ Xλ2
g := sup e−λ(x+y) g(x, y) .
(2.2)
x,y∈X
We have
Lemma 2.2 The space (Xλ2 , · ), where · is defined by (2.2), is a linear normed
space.
2.2 Quadratic Difference Operator in Xλ Spaces
We define the quadratic difference operator Q(f ) by
Q(f )(x, y) := f (x + y) + f (x − y) − 2f (x) − 2f (y)
(2.3)
for x, y ∈ X, f ∈ Xλ .
Then we have
Theorem 2.1 The quadratic difference operator Q : Xλ → Xλ2 , given by the formula (2.3), is a linear bounded operator satisfying the inequality
Q(f ) ≤ 6f , f ∈ Xλ .
(2.4)
Proof First, we shall verify that if f ∈ Xλ , then Q(f ) ∈ Xλ2 . We have
Q(f )(x, y) = f (x + y) + f (x − y) − 2f (x) − 2f (y)
≤ Mf eλ(x+y) + Mf eλ(x−y) + 2Mf eλ(x) + 2Mf eλ(y)
≤ 6Mf eλ(x+y) ,
thus Q(f ) ∈ Xλ2 as claimed.
2 Quadratic Operators and Quadratic Functional Equation
15
Clearly, Q is linear. For f ∈ Xλ , we obtain
Q(f ) = sup e−λ(x+y) f (x + y) + f (x − y) − 2f (x) − 2f (y)
x,y∈X
≤ sup e−λ(x+y) f (x + y) + sup e−λ(x+y) f (x − y)
x,y∈X
x,y∈X
+ 2 sup e−λ(x+y) f (x) + 2 sup e−λ(x+y) f (y)
x∈X
y∈X
= f + f + 2f + 2f = 6f .
Therefore,
Q(f ) ≤ 6f ,
f ∈ Xλ ,
which concludes the proof.
Under some additional assumptions, we can prove some further results. In fact,
we have
Theorem 2.2 Let R ⊂ X, R ⊂ Y and x = |x| for x ∈ R. Then
Q = 6.
(2.5)
Proof Let {xn } be a strictly decreasing sequence of positive numbers such that
lim xn = 0.
n→∞
Let us define for n ∈ N a function fn by
⎧ 2λx
−e n ,
⎪
⎪
⎪
⎨e2λxn ,
fn (x) := 2λx
⎪
e n,
⎪
⎪
⎩
0,
Clearly, we have
x = xn ,
x = 2xn ,
x = 0,
otherwise.
fn (x) ≤ e2λxn eλx ,
x ∈ X,
so fn ∈ Xλ for all n ∈ N. Moreover,
⎧ 2λx
e n,
⎪
⎪
⎪
⎨e2λxn ,
e−λx fn (x) =
⎪
1,
⎪
⎪
⎩
0,
x = 0,
x = xn ,
x = 2xn ,
otherwise.
16
M. Adam and S. Czerwik
Because the sequence {xn } is a decreasing sequence of positive numbers convergent
to zero, we obtain that fn = e2λxn for all n ∈ N. We also have
Q(fn ) = sup e−λ(x+y) f (x + y) + f (x − y) − 2f (x) − 2f (y)
x,y∈X
≥ e−λxn fn (2xn ) + fn (0) − 4fn (xn ) = e−λxn · 6eλxn = 6.
Thus Q(fn ) ≥ 6 for n ∈ N. We also know from (2.4) that Q ≤ 6. Suppose on
the contrary that Q < 6. Then there exists ε > 0 such that
Q(fn ) ≤ (6 − ε)fn , fn ∈ Xλ .
On the other hand, we have for fn ∈ Xλ
6 ≤ Q(fn ) ≤ (6 − ε)e2λxn .
Taking into account that xn → 0 as n → ∞, we get 6 ≤ 6 − ε, where ε > 0, which is
impossible. Thus we obtain eventually that Q = 6, and the proof is complete. For further information on new results concerning the quadratic difference operator on other spaces, see also the papers [9, 11, 12].
2.3 D’Alembert and Lobaczewski Difference Operators in Xλ
Spaces
In this section, we shall recall the definition of the quadratic bounded operator. The
Lobaczewski difference operator is an interesting example of a quadratic operator.
Here we shall present the ideas and main results obtained by S. Czerwik and
K. Król in [13].
Let X and Y be linear spaces over a field K.
Definition 2.3 An operator Q : X → Y is called quadratic if it satisfies the following equations
Q(x + y) + Q(x − y) = 2Q(x) + 2Q(y),
Q(kx) = k Q(x),
2
x, y ∈ X,
x ∈ X, k ∈ K.
(2.6)
(2.7)
Definition 2.4 A quadratic operator Q : X → Y , where X, Y are linear normed
spaces over K, is called bounded if there exists an M ≥ 0 such that
Q(x) ≤ Mx2 , x ∈ X.
(2.8)
A norm of a quadratic bounded operator Q : X → Y is defined by
Q := sup Q(x) : x ≤ 1 .
x∈X
(2.9)
2 Quadratic Operators and Quadratic Functional Equation
17
By BQ (X, Y ) we denote the space of all bounded quadratic operators. It is easy
to prove that BQ (X, Y ) with the norm given by (2.9) is a linear normed space.
Let C denote the set of all complex numbers. For a set X, a symbol CX denotes
the set of all functions f : X → C.
Definition 2.5 For a linear space X, the Lobaczewski difference operator L : CX →
2
CX is defined by
2 x +y
L(f )(x, y) := f
− f (x)f (y), x, y ∈ X.
(2.10)
2
One can verify that we have
2
Remark 2.1 The Lobaczewski difference operator L : CX → CX defined by (2.10)
is a quadratic operator.
We can also prove that the Lobaczewski operator L : Xλ → Xλ2 , where Y = C, is
a quadratic bounded operator. We have even more (see [13]).
Theorem 2.3 Let Y = C. The Lobaczewski difference operator defined by (2.10)
belongs to BQ (Xλ , Xλ2 ), and for all f ∈ Xλ we have
L(f ) ≤ 2f 2 .
(2.11)
Under some additional assumptions, we can find the norm of L. In fact, the following is true.
Theorem 2.4 Let R+ ⊂ X, Y = C, and x = |x| for all x ∈ R+ . Then
L = 2,
(2.12)
where L is given by (2.10).
The proof, similar to the proof of Theorem 2.2, can be found in [13].
Now we shall present results about the d’Alembert difference operator.
Definition 2.6 We denote by BLQ the space
BLQ (X, Y ) := T ∈ Y X : ∃L ∈ B(X, Y ) and ∃Q ∈ BQ (X, Y )
such that T = L + Q .
Here, of course, B(X, Y ) stands for the space of linear bounded operators from
X to Y .
For T = L + Q ∈ BLQ (X, Y ), we define
T := L + Q.
18
M. Adam and S. Czerwik
We say that such an operator T is a bounded linear-quadratic operator.
Definition 2.7 Let X be a linear space. The d’Alembert difference operator A :
2
CX → CX is defined by
A(f )(x, y) := f (x + y) + f (x − y) − 2f (x)f (y),
x, y ∈ X.
(2.13)
In the sequel, we present the following
Theorem 2.5 ([13]) Let Y = C and X be a normed space. The d’Alembert difference operator A : Xλ → Xλ2 defined by (2.13) belongs to BLQ (Xλ , Xλ2 ), and for all
f ∈ Xλ we have
A(f ) ≤ 2f + 2f 2 .
Proof On account of (2.13), we get A = LA + QA , where the linear operator LA :
Xλ → Xλ2 and the quadratic operator QA : Xλ → Xλ2 are given by
LA (f )(x, y) := f (x + y) + f (x − y),
QA (f )(x, y) := −2f (x)f (y).
Now, for any f ∈ Xλ we obtain successively
LA (f ) = sup e−λ(x+y) f (x + y) + f (x − y)
x,y∈X
≤ sup e−λ(x+y) f (x + y) + sup e−λ(x+y) f (x − y)
x,y∈X
≤ sup e
x,y∈X
f (x + y) + sup e−λx+y f (x − y) = 2f .
−λx+y x,y∈X
x,y∈X
Therefore, LA ∈ B(Xλ , Xλ2 ). We shall now prove that QA is bounded and
QA = 2. Indeed, for f ∈ Xλ we get
QA (f ) = sup e−λx+y 2f (x)f (y)
x,y∈X
= 2 sup e−λx f (x) · sup e−λy f (y) = 2f 2 .
x∈X
y∈X
Thus QA ∈ BQ (Xλ , Xλ2 ) and QA = 2. Since A = LA + QA , we get that A ∈
BLQ (Xλ , Xλ2 ) and
A(f ) = LA (f ) + QA (f ) ≤ LA (f ) + QA (f ) ≤ 2f + 2f 2 ,
as claimed.
Under additional assumptions, one can compute the norm of A. Namely, we have
2 Quadratic Operators and Quadratic Functional Equation
19
Theorem 2.6 Let X be a linear normed space, R+ ⊂ X, Y = C, and x = |x| for
x ∈ R+ . Then
A = 4.
The proof, similar to the proof of Theorem 2.2, can be found in [13].
2.4 Quadratic Functional Equation and Functional Equations
for Quadratic Differences
At first, we shall give the formula for the general solution of the generalized
quadratic functional equation on a group. The result is due to K. Dłutek (see [7]).
Theorem 2.7 Let G1 and G2 be groups with division by two. Let A, B, C, D : G1 →
G2 satisfy the equation
A(x) + B(y) = C(x + y) + D(x − y),
x, y ∈ G1 .
Then there exist a quadratic function K : G1 → G2 (i.e., a function satisfying the
equation (2.6)), additive functions E, F : G1 → G2 and constants S1 , S2 , S3 , S4 ∈
G2 such that
A(x) = 2K(x) + E(x) + F (x) + S1 ,
B(x) = 2K(x) + E(x) − F (x) + S2 ,
C(x) = K(x) + E(x) + S3 ,
D(x) = K(x) + F (x) + S4
for all x ∈ G1 and S1 + S2 = S3 + S4 .
Now we shall state the result concerning the properties of the quadratic difference
operator Q on LP -spaces; for more details, see [11].
Theorem 2.8 Let (G, Σ, μ) be a complete measurable Abelian group, μ(G) < ∞
and let (E, · ) be a Banach space. If 1 ≤ p ≤ ∞, then the quadratic difference
operator
Q : LPμ (G, E) → LPμ×μ (G × G, E)
given by (2.3) is linear, continuous, and invertible. Moreover, the inverse operator
Q−1 defined for h ∈ Q[LPμ (G, E)] is continuous and has the form
−1
Q
−1
h(·) = 2μ(G)
h(x, ·) dμ(x).
G
20
M. Adam and S. Czerwik
For some problems, particularly for the problem of Ulam–Hyers–Rassias stability of functional equations, functional equations for quadratic differences are very
useful (see [3, 7, 8]). Let us present a few such equations, which we will need in the
proof of Theorem 2.11.
Theorem 2.9 Let X, Y be Abelian groups and f : X → Y be a function. Then Q(f )
given by formula (2.3) satisfies the following functional equations
Q(f )(x + y, s + t) + Q(f )(x − y, s − t) + 2Q(f )(x, y) + 2Q(f )(s, t)
= Q(f )(x + s, y + t) + Q(f )(x − s, y − t) + 2Q(f )(x, s)
+ 2Q(f )(y, t),
(2.14)
Q(f )(x + y, s) + Q(f )(x − y, s) + 2Q(f )(x, y)
= Q(f )(x + s, y) + Q(f )(x − s, y) + 2Q(f )(x, s),
(2.15)
Q(f )(x + y, t) + Q(f )(x − y, t) + 2Q(f )(x, y)
= Q(f )(x, y + t) + Q(f )(x, y − t) + 2Q(f )(y, t)
(2.16)
for all x, y, s, t ∈ X.
There are also interesting partial differential equations for quadratic differences
(see [3, 7, 8]). Let X and Y be normed spaces. The space of all functions f : X → Y
that are n-times differentiable will be denoted by D n (X, Y ). By ∂kn f , k = 1, 2, we
denote, as usual, the nth partial derivative of f : X × X → Y with respect to the kth
variable.
Theorem 2.10 Let f : X → Y be a function such that Q(f ) ∈ D 2 (X × X, Y ). Then
we have
∂22 Q(f ) (x + y, 0) + ∂22 Q(f ) (x − y, 0)
(2.17)
= 2∂12 Q(f ) (x, y) + 2∂22 Q(f ) (x, 0),
∂22 Q(f ) (x + y, 0) + ∂22 Q(f ) (x − y, 0)
(2.18)
= 2∂22 Q(f ) (x, y) + 2∂22 Q(f ) (y, 0),
2
2∂12
Q(f ) (x, y) = ∂22 Q(f ) (x + y, 0) − ∂22 Q(f ) (x − y, 0) (2.19)
for all x, y ∈ X.
From Theorems 2.9 and 2.10, we easily obtain the following corollary.
Corollary 2.1 Let f : X → Y be a function such that Q(f ) ∈ D 2 (X × X, Y ). Then
we have
∂1 Q(f ) (0, 0) = 0,
(2.20)
2 Quadratic Operators and Quadratic Functional Equation
2
Q(f ) (0, 0) = 0,
∂12
∂22 Q(f ) (0, 0) = 0.
Moreover, for all x ∈ X we have
∂2 Q(f ) (x, 0) = ∂2 Q(f ) (0, 0).
21
(2.21)
(2.22)
2.5 Double Quadratic Difference Property
In 1940, S.M. Ulam posed the following problem (cf. [29]):
We are given a group (X, +) and a metric group (Y, +, d). Given ε > 0, does
there exist a δ > 0 such that if f : X → Y satisfies the inequality
d f (x + y), f (x) + f (y) < δ for all x, y ∈ X,
then a homomorphism A : X → Y exists with
d f (x), A(x) < ε for all x ∈ X?
One can ask a similar question for other important functional equations. The first
partial solution of this problem was given by D.H. Hyers [16] under the assumption
that X and Y are Banach spaces. In 1978, Themistocles M. Rassias extended the
theorem of Hyers by considering an unbounded Cauchy difference (see [23]). During the last decades, the stability problems of various functional equations have been
extensively investigated by many authors (see, e.g., [1, 2, 7, 8, 14, 15, 17, 18, 24–
27]).
Assume that X and Y are normed spaces. For a function f : X → Y , we put
f sup := sup f (x).
x∈X
For the quadratic difference, the stability problem can be reformulated as follows.
Let ε > 0 be given. Does there exist a δ > 0 such that if f : X → Y satisfies
Q(f ) < δ,
sup
then there exists a quadratic function K : X → Y with
f − Ksup < ε?
We can consider Ulam’s problem for different norms. In this paper, we are going
to prove the stability of the quadratic functional equation in the class of differentiable functions. The same problem for the Cauchy type functional equations was
solved by J. Tabor and J. Tabor in [28].
Let X and Y be a real normed space and a real Banach space, respectively. By
N0 , N, R we denote the sets of all nonnegative integers, positive integers, and real
22
M. Adam and S. Czerwik
numbers, respectively. Let f : X → Y be an n-times Fréchet differentiable function. By D n f , n ∈ N, we denote the nth derivative of f , and D 0 f stands for f . By
C n (X, Y ) we denote the space of n-times continuously differentiable functions and
by BC n (X, Y ) the subspace of C n (X, Y ) consisting of bounded functions. Moreover, C 0 (X, Y ) and C ∞ (X, Y ) stand for the space of continuous functions and the
space of infinitely many times continuously differentiable functions, respectively.
Following an idea of J. Tabor and J. Tabor [28], we assume that we are given a
norm in X × X such that (x1 , x2 ) is a function of x1 and x2 , and the following
condition is satisfied
(x, 0) = (0, x) = x, x ∈ X.
Let i1 : X → X × X, i2 : X → X × X be injections defined by
i1 (x) := (x, 0),
x ∈ X,
i2 (y) := (0, y),
y ∈ X.
Let L : X × X → X be a bounded linear mapping. It follows directly from the assumed conditions on the norm in X × X that
L ◦ i1 ≤ Li1 = L,
L ◦ i2 ≤ Li2 = L.
Therefore, if F : X × X → Y is n-times differentiable for n ∈ N, then
∂1 F (x, y) = DF (x, y) ◦ i1 ≤ DF (x, y),
∂2 F (x, y) = DF (x, y) ◦ i2 ≤ DF (x, y)
and
∂1i−2 ∂22 F (x, y) ≤ D i F (x, y),
(2.23)
∂12 ∂2i−2 F (x, y) ≤ D i F (x, y)
(2.24)
for all x, y ∈ X and i = 2, 3, . . . , n.
Let n ∈ N and let f : X → Y be n-times differentiable. Then Q(f ) is also ntimes differentiable, and by (2.24) we have
Df (x + y) − Df (x − y) − 2Df (y) ≤ D Q(f ) (x, y),
(2.25)
2
D f (x + y) + D 2 f (x − y) − 2D 2 f (y) ≤ D 2 Q(f ) (x, y) (2.26)
for all x, y ∈ X. Moreover, for n ≥ 3, we obtain from (2.24)
i
D f (x + y) + D i f (x − y) ≤ D i Q(f ) (x, y)
for all x, y ∈ X and i = 3, 4, . . . , n.
(2.27)
2 Quadratic Operators and Quadratic Functional Equation
23
We will prove that the class C n (R, Y ) has the so-called double quadratic difference property, i.e., if f : R → Y is such a function that Q(f ) ∈ C n (R × R, Y ),
then there exists exactly one quadratic function K0 : R → Y such that f − K0 ∈
C n (R, Y ) (see also [3]). The problem of the double difference property for the
Cauchy difference C(f )(x, y) := f (x + y) − f (x) − f (y) ∈ C n (X × X, Y ) has
been investigated in [28]. For more details about the double difference property, the
reader is referred to [19].
Lemma 2.3 (See also [3]) Let f : X → Y be a function such that Q(f ) ∈ C 2 (X ×
X, Y ). Then K0 : X → Y given by the formula
1 K0 (x) = f (x) − f (0) + ∂2 Q(f ) (0, 0)(x)
2
1 t
1
−
∂ 2 Q(f ) (ux, 0) x 2 du dt,
2 0 0 2
x∈X
(2.28)
is a quadratic function.
Proof Let f1 (x) := f (x) − f (0) for all x ∈ X. Then Q(f1 ) = Q(f ) + 2f (0) ∈
C n (X × X, Y ) and Q(f1 )(0, 0) = 0. Moreover, ∂2 (Q(f1 )) = ∂2 (Q(f )) and
∂22 (Q(f1 )) = ∂22 (Q(f )). Let us fix arbitrary x, y ∈ X and consider a function
ϕ(t) := Q(f1 )(tx, ty),
t ∈ R.
Obviously, ϕ ∈ C 2 (R, Y ). Then we have
Dϕ(t) = ∂1 Q(f1 ) (tx, ty)(x) + ∂2 Q(f1 ) (tx, ty)(y),
t ∈ R.
Hence and from (2.20), we get
Dϕ(0) = ∂2 Q(f1 ) (0, 0)(y).
Therefore, we obtain
1
Q(f1 )(x, y) = ϕ(1) − ϕ(0) =
1 t
=
0
=
0
1 t
Dϕ(t) dt =
0
D 2 ϕ(u) du dt + Dϕ(0)
0
D 2 Q(f1 ) (ux, uy)(x, y) du dt + ∂2 Q(f1 ) (0, 0)(y)
0
1 t
0
0
+2
∂12 Q(f1 ) (ux, uy) x 2 du dt
1 t
0
0
1 t
+
0
0
2
∂12
Q(f1 ) (ux, uy)(xy) du dt
∂22 Q(f1 ) (ux, uy) y 2 du dt + ∂2 Q(f1 ) (0, 0)(y).
24
M. Adam and S. Czerwik
Thus
Q(f1 )(x, y) =
1 t
0
0
+2
+
0
∂12 Q(f1 ) (ux, uy) x 2 du dt
1 t
0
0
1 t
0
2
Q(f1 ) (ux, uy)(xy) du dt
∂12
∂22 Q(f1 ) (ux, uy) y 2 du dt
+ ∂2 Q(f1 ) (0, 0)(y),
x, y ∈ X.
(2.29)
We define the function K0 : X → Y by the formula
1 K0 (x) := f1 (x) + ∂2 Q(f1 ) (0, 0)(x)
2
1 1 t 2
∂2 Q(f1 ) (ux, 0) x 2 du dt,
−
2 0 0
x ∈ X.
We show that K0 is a quadratic function. By making use of (2.17), (2.18), (2.19),
and (2.29), we obtain for all x, y ∈ X
K0 (x + y) + K0 (x − y) − 2K0 (x) − 2K0 (y)
= Q(f1 )(x, y) − ∂2 Q(f1 ) (0, 0)(y)
1 1 t 2
−
∂2 Q(f1 ) (ux + uy, 0)(x + y)2 du dt
2 0 0
1 1 t 2
−
∂2 Q(f1 ) (ux − uy, 0)(x − y)2 du dt
2 0 0
1 t
+
∂22 Q(f1 ) (ux, 0) x 2 du dt
0
+
=
0
1 t
0
0
1 t
0
0
∂12 Q(f1 ) (ux, uy) x 2 du dt
1 t
+2
0
0
−
1
2
0
1 t
+
∂22 Q(f1 ) (uy, 0) y 2 du dt
0
0
2
∂12
Q(f1 ) (ux, uy)(xy) du dt
∂22 Q(f1 ) (ux, uy) y 2 du dt
1 t
0
∂22 Q(f1 ) (ux + uy, 0) x 2 du dt
2 Quadratic Operators and Quadratic Functional Equation
1 t
−
0
−
−
+
1
2
0
0
0
∂22 Q(f1 ) (ux + uy, 0)(xy) du dt
1 t
0
1 t
1
2 0
1
0
t
0
25
∂22 Q(f1 ) (ux + uy, 0) y 2 du dt
∂22 Q(f1 ) (ux − uy, 0) x 2 du dt
∂22 Q(f1 ) (ux − uy, 0)(xy) du dt
1 1 t 2
∂2 Q(f1 ) (ux − uy, 0) y 2 du dt
2 0 0
1 t
+
∂22 Q(f1 ) (ux, 0) x 2 du dt
−
0
+
=
0
0
1 t
0
∂22 Q(f1 ) (uy, 0) y 2 du dt
1 t
1 ∂12 Q(f1 ) (ux, uy) − ∂22 Q(f1 ) (ux + uy, 0)
2
0
0
1 2
2
− ∂2 Q(f1 ) (ux − uy, 0) + ∂2 Q(f1 ) (ux, 0) x 2 du dt
2
1 t
2
2∂12 Q(f1 ) (ux, uy) − ∂22 Q(f1 ) (ux + uy, 0)
+
0
0
Q(f1 ) (ux − uy, 0) (xy) du dt
1 t
1 ∂22 Q(f1 ) (ux, uy) − ∂22 Q(f1 ) (ux + uy, 0)
+
2
0
0
1
− ∂22 Q(f1 ) (ux − uy, 0) + ∂22 Q(f1 ) (uy, 0) y 2 du dt = 0.
2
+ ∂22
Therefore, K0 is a quadratic function, which completes the proof.
Theorem 2.11 Let n ≥ 2 be a fixed positive integer and let f : R → Y be a function such that Q(f ) ∈ C n (R × R, Y ). Then there exists a unique quadratic function
K0 : R → Y such that f − K0 ∈ C n (R, Y ) and D 2 (f − K0 )(0) = 0. Moreover, we
have for all x ∈ R
1 1 x 2
D(f − K0 )(x) =
∂2 Q(f ) (s, 0) ds − ∂2 Q(f ) (0, 0),
2 0
2
1
D 2 (f − K0 )(x) = ∂22 Q(f ) (x, 0),
2
26
M. Adam and S. Czerwik
k
D (f − K0 )(x) ≤ 1 D k Q(f ) (x, 0),
2
k ∈ N\{1}, k ≤ n.
Proof Let f1 (x) := f (x) − f (0) for all x ∈ R. On account of Lemma 2.3, there
exists a quadratic function K0 given by (2.28). Now we prove that f − K0 is a
differentiable function. Fix arbitrary x, h ∈ R, h = 0. Then we get
1
f1 (x + h) − K0 (x + h) − f1 (x) − K0 (x)
h
1 1 1 t 2
=
∂ Q(f1 ) u(x + h), 0 (x + h)2 du dt
h 2 0 0 2
1 − ∂2 Q(f1 ) (0, 0)(x + h)
2
2
1 1 1 t 2
−
∂ Q(f1 ) (ux, 0) x du dt + ∂2 Q(f1 ) (0, 0)(x)
2 0 0 2
2
x+h v
1
=
∂22 Q(f1 ) (s, 0) ds dv
2h 0
0
x v
∂22 Q(f1 ) (s, 0) ds dv − ∂2 Q(f1 ) (0, 0)(h)
−
0
=
1
2h
x
0
x+h v
0
∂22 Q(f1 ) (s, 0) ds dv − ∂2 Q(f1 ) (0, 0)(h)
1
∂22 Q(f1 ) (s, 0)(h) ds dt − ∂2 Q(f1 ) (0, 0)(h)
2h 0 0
1 x+th
1
∂22 Q(f1 ) (s, 0) ds dt − ∂2 Q(f1 ) (0, 0)
=
2 0 0
1 1 x 2
∂ Q(f1 ) (s, 0) ds dt
−→
2 0 0 2
1 1 x 2
1 − ∂2 Q(f1 ) (0, 0) =
∂2 Q(f1 ) (s, 0) ds − ∂2 Q(f1 ) (0, 0)
2
2 0
2
=
1 x+th
for h → 0. Hence the function f − K0 = f1 − K0 + f (0) is differentiable at every
x ∈ R, ∂2 (Q(f1 )) = ∂2 (Q(f )), ∂22 (Q(f1 )) = ∂22 (Q(f )) and
D(f − K0 )(x) =
1
2
0
x
1 ∂22 Q(f ) (s, 0) ds − ∂2 Q(f ) (0, 0),
2
x ∈ R. (2.30)
Moreover, since the function f − K0 is differentiable at every x ∈ R, then there
exists also the second difference derivative which is equal to the second derivative
2 Quadratic Operators and Quadratic Functional Equation
27
(see [21]). We show that
1 D 2 (f − K0 )(x) = ∂22 Q(f ) (x, 0),
2
x ∈ R.
Fix arbitrary x, h ∈ R and let a function ψ : R → Y be given by
ψ(t) := Q(f1 )(x, th),
t ∈ R.
Then ψ ∈ C n (R, Y ) and we have
Q(f1 )(x, h) = ψ(1) − ψ(0)
1 t
∂22 Q(f1 ) (x, uh) h2 du dt
=
0
0
+ ∂2 Q(f1 ) (x, 0)(h),
x, h ∈ R.
From (2.22) we get
∂2 Q(f1 ) (x, 0) = ∂2 Q(f1 ) (0, 0),
x ∈ R,
hence
Q(f1 )(x, h) =
0
1 t
0
∂22 Q(f1 ) (x, uh) h2 du dt
+ ∂2 Q(f1 ) (0, 0)(h),
x, h ∈ R.
(2.31)
Next, from (2.18) we obtain
0
1 t
0
∂22 Q(f1 ) (x + uh, 0) h2 du dt
1 t
+
=2
0
0
1 t
0
0
+2
0
∂22 Q(f1 ) (x − uh, 0) h2 du dt
∂22 Q(f1 ) (x, uh) h2 du dt
1 t
0
∂22 Q(f1 ) (uh, 0) h2 du dt,
x, h ∈ R.
(2.32)
Therefore, using (2.31) and (2.32), for each fixed x, h ∈ R we have
f1 (x + h) − K0 (x + h) − 2f1 (x) + 2K0 (x) + f1 (x − h) − K0 (x − h)
2 1 2
− ∂2 Q(f1 ) (x, 0) h 2
28
M. Adam and S. Czerwik
2 1 2
= Q(f1 )(x, h) + 2f1 (h) − 2K0 (h) − ∂2 Q(f1 ) (x, 0) h 2
1 t
=
∂22 Q(f1 ) (x, uh) h2 du dt + ∂2 Q(f1 ) (0, 0)(h)
0
+
0
1 t
0
0
0
0
∂22 Q(f1 ) (uh, 0) h2 du dt
2 1 2
− ∂2 Q(f1 ) (0, 0)(h) − ∂2 Q(f1 ) (x, 0) h 2
1 t
1
=
∂22 Q(f1 ) (x + uh, 0) h2 du dt
2
0
0
1 t
1
∂22 Q(f1 ) (x − uh, 0) h2 du dt
+
2 0 0
1 t
2
2
−
∂2 Q(f1 ) (x, 0) h du dt =
1 t1
1 ∂22 Q(f1 ) (x + uh, 0) + ∂22 Q(f1 ) (x − uh, 0)
2
0
0 2
− ∂22 Q(f1 ) (x, 0) h2 du dt 1 2
≤ h2 sup 2 ∂2 Q(f1 ) (x + uh, 0)
u∈[0,1]
1 2
2
+ ∂2 Q(f1 ) (x − uh, 0) − ∂2 Q(f1 ) (x, 0)
.
2
Since ∂22 (Q(f1 )) = ∂22 (Q(f )) is a continuous function,
1 1 2
∂2 Q(f1 ) (x + uh, 0) + ∂22 Q(f1 ) (x − uh, 0) − ∂22 Q(f1 ) (x, 0) → 0
2
2
for h → 0. Hence we get
1 D 2 (f − K0 )(x) = ∂22 Q(f ) (x, 0),
2
x ∈ R.
(2.33)
Since ∂22 (Q(f ))(x, 0) ∈ C n−2 (R × R, Y ), one has ∂22 (Q(f ))(x, 0) ∈ C n−2 (R, Y ).
Finally, D 2 (f −K0 ) ∈ C n−2 (R, Y ), i.e., f −K0 ∈ C n (R, Y ). Moreover, from (2.21)
we also have
1 D 2 (f − K0 )(0) = ∂22 Q(f ) (0, 0) = 0.
2
To prove the uniqueness of K0 , consider quadratic functions K1 , K2 : R → Y
such that f − K1 , f − K2 ∈ C n (R, Y ), D 2 (f − K1 )(0) = D 2 (f − K2 )(0) = 0 and
2 Quadratic Operators and Quadratic Functional Equation
29
conditions (2.30), (2.33) hold. Then K1 − K2 is a quadratic function and K1 − K2 ∈
C n (R, Y ). Therefore, for every x ∈ R, we have
D 2 (K1 − K2 )(x) = D 2 (f − K2 ) − (f − K1 ) (x)
= D 2 (f − K2 )(x) − D 2 (f − K1 )(x) = 0.
Since D 2 (K1 − K2 ) = 0, D(K1 − K2 )(x) is a constant function for every x ∈ R.
But
D(K1 − K2 )(0) = D(f − K2 )(0) − D(f − K1 )(0) = 0,
so D(K1 − K2 ) = 0. Analogously, we have that (K1 − K2 )(x) is a constant function
for every x ∈ R; and since (K1 − K2 )(0) = 0, it yields that K1 = K2 .
It remains to prove that
k
D (f − K0 )(x) ≤ 1 D k Q(f ) (x, 0),
2
k ∈ N\{1}, k ≤ n
(2.34)
for every x ∈ R. Let g := f − K0 . Then g ∈ C n (R, Y ), and consequently we have
Q(g) = Q(f ) ∈ C n (R × R, Y ). Making use of (2.26) and the fact that D 2 g(0) = 0,
we obtain
2
D g(x) = D 2 g(x) − D 2 g(0) ≤ 1 D 2 Q(g) (x, 0),
2
x ∈ R,
which proves (2.34) for k = 2. For 3 ≤ k ≤ n, k ∈ N, condition (2.34) follows directly from (2.27), which completes the proof.
Corollary 2.2 ([3]) Under the assumptions of Theorem 2.11, we have
k
D (f − K0 )(0) ≤ 1 D k Q(f ) (0, 0), k ∈ N0 \{1}, k ≤ n,
2
k
D (f − K0 ) ≤ 1 D k Q(f ) , k ∈ N\{1}, k ≤ n.
sup
sup
2
(2.35)
(2.36)
Proof The case k = 0 in (2.35) is trivial because obviously f (0) = − 12 Q(f )(0, 0).
From (2.34) we obtain (2.35) for k ≥ 2 and (2.36). The proof is completed.
Remark 2.2 Let the assumptions of Theorem 2.11 be satisfied and let
∂2 (Q(f ))(0, 0) = 0. Then the inequality (2.34) (and consequently (2.35) and (2.36))
also holds for k = 1.
Proof If ∂2 (Q(f ))(0, 0) = 0, then from (2.30) we obtain D(f − K0 )(0) = 0. Let
g := f − K0 . Hence g ∈ C n (R, Y ), Q(g) = Q(f ) ∈ C n (R × R, Y ), and C(g) ∈
C n (R × R, Y ). Therefore, on account of (2.23), we get
Dg(x + y) − Dg(y) ≤ D C(g) (x, y), x, y ∈ R.
(2.37)
30
M. Adam and S. Czerwik
One can easily check that for any function h : R → Y the following equality holds
2C(h)(x, y) + 2C(h)(x, −y) = Q(h)(x, y) + Q(h)(x, −y),
x, y ∈ R,
where C(f ) denotes the Cauchy difference. Then, in particular, for a function g we
obtain
1 1 D C(g) (x, 0) = D Q(g) (x, 0) = D Q(f ) (x, 0),
2
2
x ∈ R.
Therefore, by virtue of (2.37) with y = 0, from the above equality and the fact that
Dg(0) = 0, we have
Dg(x) = Dg(x) − Dg(0) ≤ D C(g) (x, 0) = 1 D Q(g) (x, 0),
2
x ∈ R,
which proves the inequality (2.34) for k = 1.
It is still an open problem to prove that the function f − K0 which occurs in
Theorem 2.11 is differentiable for every x ∈ X, where X denotes a real normed
space.
Corollary 2.3 ([3]) The quadratic function K0 : R → Y occurring in Lemma 2.3
and Theorem 2.11 can be defined by the formula
x
x
1
+f −
− 2f (0) , x ∈ R.
(2.38)
K0 (x) = lim n2 f
2 n→∞
n
n
Theorem 2.11 states, in particular, that the class of infinitely many times differentiable functions has the double quadratic difference property. We may show that
the class of analytic functions also has this property.
Corollary 2.4 ([3]) Let f : R → Y be a function such that Q(f ) is analytic. Then
there exists exactly one quadratic function K : R → Y such that f − K is analytic
and D 2 (f − K)(0) = 0.
Now we give some auxiliary results which will be used in the sequel.
Lemma 2.4 ([5]) Let (G, +) be an Abelian group. If a function f : G → Y satisfies
the inequality
f (x + y) + f (x − y) − 2f (x) − 2f (y) ≤ ε, x, y ∈ G
for some ε > 0, then there exists a unique quadratic function K : G → Y such that
f (x) − K(x) ≤ 1 ε,
2
x ∈ G.
2 Quadratic Operators and Quadratic Functional Equation
31
Moreover, the function K is given by the formula
f (2n x)
,
n→∞ 22n
K(x) = lim
x ∈ G.
In [6], S. Czerwik provided a generalization of the above result and also proved
that if a function R t → f (tx) is continuous for each fixed x ∈ E, where E denotes a real normed space, then K(tx) = t 2 K(x) for all t ∈ R and x ∈ E.
The following lemma is some kind of an analogue to the Mean Value Theorem
for real valued functions.
Lemma 2.5 ([20]) Let a mapping T : B → Y , B ⊂ X, where B is an open set, be
two times Fréchet differentiable. Let x, h ∈ B, and let for every 0 ≤ α ≤ 1, (x +
αh) ∈ B. Then
T (x + h) − T (x) − DT (x)h ≤ 1 h2 sup D 2 T (x + αh).
2
0<α<1
Similarly as for the case of Euler’s Theorem for positive homogeneous functions
(see [22]), one can prove the following lemma.
Lemma 2.6 Let T : R → Y be a homogeneous function of degree 2 such that T ∈
C 2 (R, Y ). Then
1
T (x) = x 2 D 2 T (x),
2
x ∈ R.
(2.39)
Proof Since T is a homogeneous function of degree 2, then
T (αx) = α 2 T (x),
x, α ∈ R.
Let us fix an arbitrary x ∈ R. Differentiating both sides of the above equality with
respect to α, we obtain
D 2 T (αx)x 2 = 2T (x),
α ∈ R.
Since x ∈ R was chosen arbitrarily, for α = 1 we get the equality (2.39), which
completes the proof.
Lemma 2.7 Let K : R → Y be a quadratic function and let f : R → Y be a mapping such that f ∈ C 2 (R, Y ). Assume also that f − K is bounded and
(2.40)
sup D 2 f (x) − D 2 f (y) ≤ ε
(x,y)∈R×R
for some ε > 0. Then K is differentiable and
sup D 2 K(x) − D 2 f (x) ≤ ε.
x∈R
(2.41)
32
M. Adam and S. Czerwik
Proof Since f − K is bounded, then Q(f ) is also bounded and, on account of
Lemma 2.4, we have
f (2n x)
,
n→∞ 22n
K(x) = lim
x ∈ R.
Since f is a continuous at each x ∈ R, K is also continuous and it is of the form
(see also [7])
K(x) = x 2 K(1),
x ∈ R.
Thus K is differentiable. Let us fix an arbitrary y ∈ R. Applying Lemma 2.5 to the
function
1
T (x) := f (x) − x 2 D 2 f (y),
2
x∈R
and the inequality (2.40), we get for all x ∈ R
f (x) − 1 x 2 D 2 f (y) − f (0) − xDf (0) ≤ 1 |x|2 sup D 2 f (x) − D 2 f (y)
2
2
x∈R
1
≤ ε|x|2 .
2
Replacing x by 2n x and dividing both sides of the above inequality by 2n , we have
f (2n x) 1 2 2
f (0) xDf (0) 2
1
22n − 2 x D f (y) − 22n − 2n ≤ 2 ε|x| , x ∈ R.
Letting n → ∞, we conclude that
K(x) − 1 x 2 D 2 f (y) ≤ 1 ε|x|2 ,
2
2
x ∈ R.
Therefore, by virtue of Lemma 2.6, we obtain
1 2 2
x D K(x) − 1 x 2 D 2 f (y) ≤ 1 ε|x|2 ,
2
2
2
and hence
2
D K(x) − D 2 f (y) ≤ ε,
x ∈ R,
x ∈ R.
Since y was arbitrary,
sup D 2 K(x) − D 2 f (x) ≤ ε,
x∈R
which completes the proof.
2 Quadratic Operators and Quadratic Functional Equation
33
Theorem 2.12 Let n ≥ 2 be a fixed positive integer and let f : R → Y be such a
function that Q(f ) ∈ BC n (R × R, Y ). Then there exists a unique quadratic function
K∞ : R → Y such that f − K∞ ∈ BC n (R, Y ). Moreover,
k
D (f − K∞ )(0) ≤ 1 D k Q(f ) (0, 0), k ∈ N0 \{2}, k ≤ n, (2.42)
2
k
D (f − K∞ ) ≤ 1 D k Q(f ) , k ∈ N0 \{1}, k ≤ n.
(2.43)
sup
sup
2
Proof By virtue of Theorem 2.11, there exists a unique quadratic function K0 : R →
Y such that f1 := f − K0 ∈ C n (R, Y ). Then Q(f1 ) = Q(f ) ∈ BC n (R × R, Y ). It
means, in particular, that Q(f1 ) is bounded. By Lemma 2.4, there exists a unique
quadratic function K1 : R → Y such that
1
sup f1 (x) − K1 (x) ≤
sup Q(f1 )(x, y).
2 (x,y)∈R×R
x∈R
(2.44)
We put
K∞ (x) := K0 (x) + K1 (x),
x ∈ R.
Clearly, K∞ is also a quadratic function and f − K∞ = f1 − K1 ∈ C n (R, Y ). From
(2.26) we obtain
2
D f1 (x) − D 2 f1 (y) ≤ 1 sup D 2 Q(f1 ) (x, y).
2 (x,y)∈R×R
(x,y)∈R×R
sup
(2.45)
Conditions (2.44) and (2.45) mean that the functions f1 and K1 satisfy the assumptions of Lemma 2.7 with
ε=
1
sup D 2 Q(f1 ) (x, y).
2 (x,y)∈R×R
Thus K1 is differentiable and, by making use of (2.41) and (2.45), we get
sup D 2 (f − K∞ )(x) = sup D 2 (f1 − K1 )(x)
x∈R
x∈R
≤
1
sup D 2 Q(f1 ) (x, y).
2 (x,y)∈R×R
(2.46)
For k = 0, the inequality (2.42) is obvious; for k = 1, it follows from (2.25); and for
k ≥ 3, it is a trivial consequence of (2.27). Making use of (2.44), (2.46), and (2.27),
we obtain (2.43), which completes the proof.
Corollary 2.5 The quadratic function K∞ occurring in Theorem 2.12 can be defined by the formula
f (2n x)
,
n→∞ 22n
K∞ (x) = lim
x ∈ R.
34
M. Adam and S. Czerwik
Proof The formula for K∞ is a trivial consequence of the fact that the function
f − K∞ is bounded.
Comparing the formulae for K0 and K∞ , one can easily notice that these functions are usually different. We will see it in the following example.
Example Let f : R → Y be a bounded function such that f ∈ C 2 (R, Y ). Since f is
bounded, K∞ = 0. One can easily check that the following equalities hold:
∂2 Q(f ) (0, 0) = −2Df (0),
∂22 Q(f ) (x, 0) = 2D 2 f (x) − 2D 2 f (0), x ∈ R.
Therefore, by applying the formula of K0 given by (2.28), we obtain that
1
K0 (x) = D 2 f (0)x 2 ,
2
x ∈ R,
D 2 K0 (x) = D 2 f (0),
x ∈ R.
(2.47)
hence
D 2 f (0) = 0.
Thus K0 = K∞ if and only if
Clearly, one can also obtain the formula (2.47) from (2.38).
2.6 Stability
Let f : X → Y be a function such that f ∈ C n (X, Y ) for n ∈ N0 . In subspaces
of C n (X, Y ), one can consider different norms defined in terms of D i f (0),
D i f sup for i ≤ n. For example, the following norms
f :=
n−1
i
D f (0) + D n f sup
,
i=0
f :=
n
i D f sup
,
(2.48)
i=0
f := max D i f sup
i=0,...,n
are used very often. Obviously, several other norms can be introduced. We will prove
the stability result for the quadratic difference Q(f ) in a possibly general
setting.
We will use the following convention: if m, n ∈ N0 and m > n, then ni=m ai = 0.
In the sequel, we will use the following assumptions introduced in [28]. Let n ∈
N0 ∪ {∞} be fixed. In the set [0, ∞]2n+2 , we introduce the following order
(x1 , x2 , . . .) ≤ (y1 , y2 , . . .)
iff xi ≤ yi for i ∈ N, i ≤ 2n + 2.
2 Quadratic Operators and Quadratic Functional Equation
35
Let p : [0, ∞]2n+2 → [0, ∞] be any function satisfying the following conditions:
(i) p(x + y) ≤ p(x) + p(y), x, y ∈ [0, ∞]2n+2 ,
(ii) p(αx) = αp(x), x, y ∈ [0, ∞]2n+2 , α ∈ [0, ∞],
(iii) x ≤ y =⇒ p(x) ≤ p(y), x, y ∈ [0, ∞]2n+2 .
We additionally assume that 0 · ∞ = 0. From (ii) we obtain that p(0) = 0.
We define the mapping Φ : C n (X, Y ) → [0, ∞]2n+2 by the formula
Φ(f ) := f (0), f sup , Df (0), Df sup , . . .
and put
Sp (X, Y ) := f ∈ C n (X, Y ) : p Φ(f ) < ∞ .
Since p(0) = 0, Sp contains at least the zero function. It is easy to notice that Sp is
a linear space and that p ◦ Φ|Sp is a seminorm. We will denote this seminorm by
· p . The same notations we will apply for the space C n (X × X, Y ).
Now we are able to prove the main theorem of this section.
Theorem 2.13 Let f : R → Y be a function such that Q(f ) ∈ Sp (R × R, Y ) and
∂2 (Q(f ))(0, 0) = 0. We additionally assume that the function p does not depend
on the second or fourth and fifth variables. Then there exists a quadratic function
K : R → Y such that f − K ∈ Sp (R, Y ) and
1
f − Kp ≤ Q(f )p .
2
Proof Assume that Q(f ) ∈ C n (R × R, Y ). Suppose that p does not depend on the
second variable. Then
p(0, ∞, 0, . . .) = p(0, 0, 0, . . .) = 0.
By Theorem 2.11, there exists exactly one quadratic function K0 : R → Y satisfying
conditions (2.35) and (2.36). Then
Φ(f − K0 ) ≤
1 Φ Q(f ) + (0, ∞, 0, . . .) ,
2
and hence from (i), (ii), and (iii) we have
1 1 p Φ(f − K0 ) ≤ p Φ Q(f ) + (0, ∞, 0, . . .) ≤ p Φ Q(f ) ,
2
2
i.e.,
1
f − K0 p ≤ Q(f )p .
2
36
M. Adam and S. Czerwik
Suppose now that p does not depend on the fourth and fifth variables. If Q(f ) ∈
BC n (R × R, Y ), then by Theorem 2.12 there exists exactly one quadratic function
K∞ : R → Y satisfying conditions (2.42) and (2.43). Hence
Φ(f − K∞ ) ≤
1 Φ Q(f ) + (0, 0, 0, ∞, ∞, 0, . . .) ,
2
and consequently from (i), (ii), and (iii) we get
1 1 p Φ(f − K∞ ) ≤ p Φ Q(f ) + (0, 0, 0, ∞, ∞, 0, . . .) ≤ p Φ Q(f ) ,
2
2
i.e.,
1
f − K∞ p ≤ Q(f )p .
2
If Q(f ) is unbounded, then Q(f )sup = ∞. By Theorem 2.11, we can find
a quadratic function such that the conditions (2.35) and (2.36) hold. Then Φ(f −
K0 ) ≤ 12 Φ(Q(f )), and hence
1
f − K0 p ≤ Q(f )p .
2
The proof is completed.
One can easily notice that if we defined for n ∈ N0
p(x1 , x2 , . . . , x2n+2 ) :=
n
x2i−1 + x2n+2 ,
(2.49)
i=1
then we would obtain stability of the quadratic functional equation in the norm
defined by the formula (2.48).
References
1. Aczel, J.: Lectures on Functional Equations and Their Applications. Academic Press, New
York (1966)
2. Aczel, J., Dhombres, J.: Functional Equations in Several Variables. Cambridge University
Press, Cambridge (1989)
3. Adam, M., Czerwik, S.: On the double quadratic difference property. Int. J. Appl. Math. Stat.
7, 18–26 (2007)
4. Bielecki, A.: Une remarque sur la méthode de Banach–Cacciopoli–Tikhonov dans la théorie
des équations différentielles ordinaires. Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 4,
261–264 (1956)
5. Cholewa, P.W.: Remarks on the stability of functional equations. Aequ. Math. 27, 76–86
(1984)
6. Czerwik, S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Semin.
Univ. Hamb. 62, 59–64 (1992)
2 Quadratic Operators and Quadratic Functional Equation
37
7. Czerwik, S.: Functional Equations and Inequalities in Several Variables. World Scientific, New
Jersey (2002)
8. Czerwik, S. (ed.): Stability of Functional Equations of Ulam–Hyers–Rassias Type. Hadronic
Press, Palm Harbor (2003)
9. Czerwik, S., Dłutek, K.: Superstability of the equation of quadratic functionals in LP -spaces.
Aequ. Math. 63, 210–219 (2002)
10. Czerwik, S., Dłutek, K.: Cauchy and Pexider operators in some function spaces. In: Rassias,
Th.M. (ed.) Functional Equations, Inequalities and Applications, pp. 11–19. Kluwer Academic, Dordrecht (2003)
11. Czerwik, S., Dłutek, K.: Quadratic difference operators in LP -spaces. Aequ. Math. 67, 1–11
(2004)
12. Czerwik, S., Dłutek, K.: Stability of the quadratic functional equation in Lipschitz spaces. J.
Math. Anal. Appl. 293, 79–88 (2004)
13. Czerwik, S., Król, K.: The D’Alembert and Lobaczewski difference operators in Xλ spaces.
Nonlinear Funct. Anal. Appl. 13, 395–407 (2008)
14. Forti, G.L.: Hyers–Ulam stability of functional equations in several variables. Aequ. Math. 50,
143–190 (1995)
15. Ger, R.: A survey of recent results on stability of functional equations. In: Proc. of the 4th
ICFEI, Pedagogical University of Cracow, pp. 5–36 (1994)
16. Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27,
222–224 (1941)
17. Hyers, D.H., Isac, G., Rassias, Th.M.: Stability of Functional Equations in Several Variables.
Birkhäuser, Basel (1998)
18. Jung, S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Florida (2000)
19. Laczkovich, M.: Functions with measurable differences. Acta Math. Acad. Sci. Hung. 35,
217–235 (1980)
20. Luenberger, D.G.: Optimization by Vector Space Methods. PWN, Warsaw (1974) (in Polish)
21. Lusternik, L.A., Sobolew, W.I.: Elements of Functional Analysis. PWN, Warsaw (1959) (in
Polish)
22. Maurin, K.: Analysis, Part One: Elements. PWN, Warsaw (1973) (in Polish)
23. Rassias, Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc.
72(2), 297–300 (1978)
24. Rassias, Th.M.: On the stability of functional equations in Banach spaces. J. Math. Anal. Appl.
251, 264–284 (2000)
25. Rassias, Th.M.: On the stability of functional equations and a problem of Ulam. Acta Appl.
Math. 62, 23–130 (2000)
26. Rassias, Th.M. (ed.): Functional Equations and Inequalities. Kluwer Academic, Norwell
(2000)
27. Rassias, Th.M., Tabor, J. (eds.): Stability of Mapping of Hyers–Ulam Type. Hadronic Press,
Florida (1994)
28. Tabor, J., Tabor, J.: Stability of the Cauchy type equations in the class of differentiable functions. J. Approx. Theory 98(1), 167–182 (1999)
29. Ulam, S.M.: Problems in Modern Mathematics. Science Editions. Wiley, New York (1960)
http://www.springer.com/978-1-4614-3497-9